Hydrologic Design 4501

Chapter 6: Surface Water and Runoff Processes

Ardeshir Ebtehaj
University of Minnesota
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Table of Contents
Chapter 6: Surface Water and Runoff Processes 1- Watersheds and Runoff 2- Basics of Surface Runoff 3- Streamflow Hydrograph 3-1 Hydrograph Components: 3-2 Baseflow Separation: 3-3 Excess Rainfall and Direct Runoff: 3-3-1 Excess Rainfall using Streamflow Data: 3-3-2 Excess Rainfall Using Infiltration Methods: 5- Surface flow and velocity 6- Horton Laws of Stream Network

1- Watersheds and Runoff

A waterhsed, a catchment or a drainage basin, are three synonymous terms that define as a locus of all points on the Earth's surface that drain precipitation water to a single point called the wathershed outlet.
Figure 1: Left: Schematic of precipitation in a watershed flowing the precipitation water towards the outlet point. Right: A schematic of how watersheds are nested and watershed size varies depending on the basin outlet selected (right, credit: Marsh, 1998, p. 170)

2- Basics of Surface Runoff

During a rainfall event, there are two types of water storages:
Surface runoff or overland flow is defined as the portion of rainfall, snowmelt, and/or irrigation water that runs over the soil surface toward the stream network rather than infiltrating into the soil. The definition of runoff also may include the interflow and/or return flow, which together with the surface runoff make up the volume of stormwater or total runoff. The return flow is referred to the shallow subsurface lateral infiltrated flow of water that returns back to the surface.
During a precipitation event, the detention storages begin to fill out and water starts to flow over land surfaces. Overland flows join together over a hillslope and form more concentrated flows that channelize the underlying soil creating the so-called channel or stream network. Several channel networks eventually joint and drain into the main stream and generate the streamflow.
Figure 2: A schematic of the main hydrologic fluxes in streamflow.
The flow that you see in a stream or river typically consists of three main hydrologic fluxes that we briefly covered in Chapter 1. These fluxes are: overland flow, return flow of subsurface water, and baseflow as shown in Figure 4. The overland flow component can be split into two categories:
inf_all.gifinfilt_excess_flow.gifinf_saturated-)1.gif
As we discussed, the infiltration rate is the water flux that enters the soil at the surface. It is often expressed as depth of water per time, for example, 10 millimeters per hour.
If precipitation rate is much higher than the infiltration capacity, overland flow occurs immediately after the onset of precipitation. Infiltration excess is commonly observed with short-duration intense rainfall. It also occurs most often over land surfaces with high clay content or where the surface has been altered by soil compaction, urbanization, or fire.
However, when precipitation rate is less than or equal to infiltration capacity, no surface runoff occurs. For long enough low rate of precipitation, the water table may rise and produce the saturated overland flow from below. It is most common with long-duration, light-to-moderate rainfall, or because of successive precipitation and or snowmelt events at upstream. Saturation excess runoff can occur anywhere, when the soil is saturated from below. It is the most common runoff generation mechanism in humid climates with gently sloped or flat basins. The saturate overland flow and return flow often occur near the stream channels, where the groundwater depth is often shallower and it rises quickly in response to infiltration -- called groundwater ridges.
Figure 4: A schematic of overland versus saturation excess runoff.

3- Streamflow Hydrograph

The streamflow hydrograph is the time series of flow rate at a specific location on a stream. These measurements are typically captured at stream gauges, which measure flow rate based on river stage and a flow rating curve. The USGS is the primary source of stream gauge data in U.S.
For simplicity, we will categorize two different hydrographs typically analyzed based on the time scale of interest. An annual hydrograph focuses on the flow over a time period of a year at daily time steps (shown below). A storm hydrograph focuses on the flow corresponding to a precipitation event for time periods ranging from hours to weeks depending on the watershed size and storm duration.
Figure 5: An example of an annual (left) and storm (right) hydrograph for Vermillion River near Empire, MN. (Credit: USGS)
Figure 6: The main components of a streamflow in a dry period (left) and during a rainfall event (right) (from Mosley and McKerchar, 1993)

There are multiple characteristics of a basin and its river network that determine the shape of the outflow hydrograph including:
Figure 7: The effects of basin characteristics on the flood hydrograph (Masch 1984)
The dynamics of precipitation events also affects the shape of the hydrograph.
Figure 8: Effects of the shape of the rainfall hyetograph, storm size and movement on the shape of the streamflow hydrograph.

3-1 Hydrograph Components:

Hydrographs are typically divided into two components for analysis. The baseflow, which is the long-term contribution of groundwater flow to a stream hydrograph and the direct runoff, which is the portion of the hydrograph that is in response to an excess rainfall of an event and is also referred to as quick flow.
The storm hydrograph can be divided into a few segments as follows:
Figure 9: Main components of a storm hydrograph (McKerchar 1993). Note that the saturated overland flow and interflow (return flow) are the main components of the direct runoff hydrograph.

3-2 Baseflow Separation:

Three differing methods for basflow separation are:
Figure 10: Components of the streamflow hydrograph: (1-2) baseflow recession, (2-3) rising limb , (3-5) crest segment, (4) peak flow, (5-6) falling limb, and (6-7) baseflow recession. The straight line method is used for baseflow septation. At the beginning of the falling limb is an inflection point (5), where the direction of the curvature changes.
Figure 11: Fixed-based method for baseflow septation.
Figure 12: Variable slope method for baseflow septation.

3-3 Excess Rainfall and Direct Runoff:

The direct runoff determined from a hydrograph does not exactly correspond to the volume of precipitation delivered over the basin since there are losses to storages. Therefore, when predicting direct runoff from precipitation inputs, we divide the rainfall hayetograph into:
The conceptual relationships of rainfall, infiltration rate, and excess rainfall are shown in the following figures.
Figure 13: The concept of rainfall excess, which is the difference between the actual rainfall hyetograph and losses largely due to the infiltration. Generally, we either assume a constant rate (red line) or a time varying infiltration (black) rate in computation of the excess rainfall hayetograph (ERH).
Approximating excess or effective rainfall is crucial for stomrwater managment and prediction of streamflow for future rainfall events. There are different methods for determining effective rainfall depending on whether streamflow data is available or not.

3-3-1 Excess Rainfall using Streamflow Data:

To determine the excess rainfall hyetograph (ERH), the first step is to separate the baseflow from the direct runoff hydrograph (DRH). One of the most common methods for obtaining the ERH is the ϕ-Index method. This method assumes a constant rate of abstraction throughout the storm event and is calculated as follows:
where is the excess runoff depth, is the observed rainfall depth over a time interval with decreasing order , Mis the number of rainfall pulses that contribute to direct runoff, and is the constant abstraction rate.
Figure 14: A schematic representation of the ϕ-index method. The and thus are the results of direct runoff, which are obtained by subtracting the baseflow from the total observed flow.
The process consists of checking the above equation for descending values of until a positive ϕ value is achieved. Then double check that the depth of excess rainfall equals the depth of direct runoff. An example can be presented using the following information of a rainfall storm over a watershed area of 7.03 with a baseflow of 400 [cfs] (Chow et al. 1988):
Calculation:
Total amount of excess rainfall uniformly distributed over the watershed
If then we had to try and continue the process.
For characterizing the amount of excess rainfall from stream data, we can also calculate the runoff coefficient (C):
From the previous example, we can easily calculate the runoff coefficient as:
The runoff coefficient is used extensively in engineering practices to determine the peak flow rate of a storm event based on average rainfall intensity over small watersheds through the rational method:
where Q is the peak flow rate [cfs], i is the average rainfall intensity [in/hr], and A is the basin area [acres]. Note that the dimensions for the rational method do not match and it is purely an empirical relationship. In practice, this method is used to design the size of catch basins and storm sewers by calculating the peak flow rate for a certain return period of rainfall event. There are standard estimates of the runoff coefficients based on different land cover such as Lawns (C=0.05-0.35), forest (0.05-0.25), concrete streets (0.7-0.95), and etc. This topic will be covered later in the course.

3-3-2 Excess Rainfall Using Infiltration Methods:

If streamflow data is not known, then we must resort to infiltration models to calculate the excess rainfall hyetograph. It is common to use infiltration models like Green-Ampt to estimate the ERH. The key assumption is that Direct Runoff = Precipitation - Infiltration.
Recall that the Green-Ampt infiltration rate is defined as:
where denotes changes of infiltration rate in time and or are cumulative infiltration from 0 to time t. Additionally, we showed that the cumulative infiltration can be computed as follows after the ponding time:
Now for the purpose of creating the ERH, we must define cumulative infiltration after the ponding time. To that end, we showed that:
Where the cumultive infiltration at the time of ponding (), was defined as follows:
As described in the following, finding ERH using infiltration method is an iterative process.
Algorithm for determination of infiltration and ponding time under variable rainfall intensity.
WHILE ( the rainfall duration)
IF (no ponding at time t)
ELSE (ponding occurred, when infiltration rate is smaller than precipitation rate .
END
END
An example:
A rainfall hyetograph is given in col(1) and (2) of the following table. The soil is sandy loam with initial saturation . Determine the excess rainfall hyetograph.
@ t=0 [min]
@ t=10 [min]
@ t=60 [min]
Cumulative excess rainfall = cumulative rainfall - cumulative infiltration = 2.41 (col 3)-2.21 (col 6) = 0.2 [cm] (col 7)
After 140 [min], ponding ceases as
An excel file of the above calculation table is available in the class website.
Below is a plot showing the results of the example problem. Notice that all abstractions before the time of ponding (where surface runoff is generated) is labeled initial abstractions. After , everything above the green line is effective rainfall and everything below is infiltrated (losses).
Figure 15: Solution plot to Green-Ampt example.
SCS Method for runoff calculation:
The old USDA Soil Conservation Service (SCS; now called USDA-NRCS) created a runoff estimation method based on the following hypothesis shown by the components in Figure 16:
Figure 16: Schematic showing the growing contributing areas and the time of concentration
P: Total precipitation [L]
: Initial abstraction [L]
: Excess Precipitation [L]
: Water retained in the watershed [L]
S: Maximum retention storage [L]
Equation 1 can be written as follows:
and thus
From data of field experiments for small watersheds, we can assume , which leads to the following expression of excess precipitation:
Experimental results allowed parameterization of S using a curve number, which describes the retention capacity of differing landscapes:
[inches]
where CN is called the curve number
CN=100 (Impervious surfaces, e.g., pavements)
CN<100 (Natural surfaces)
Figure 17: Solution of the SCS runoff equations for normal soil moisture condition (from U.S. Department of Agricultural Soil Conservation Service 1972)
The CN depends on:
The curve number in Figure 19 are for normal antecedent moisture conditions (AMC II). The soil moisture conditions for SCS method are defined as follows:
Figure 18: Definitions of AMC based on 5-day antecedent rainfall
Therefore, correction factors are necessary for the dry and wet conditions as follows:
The CN values also depend on the soil type, which the USDA has grouped into four categories:
Figure 19: Runoff curve numbers (average watershed condition, , USDA, 1986)
The curve number in the above tables are for average watershed () and normal antecedent moisture condition (AMC II). For watersheds with several sub-catchments with different CNs, the area-averaged composite values of CN shall be computed.
Example: Compute runoff from 5 inches of rainfall on a 1000-acre watershed. Hydrologic soil type is 50% (B) and 50%(C). The watershed land use is:
for wet condition:

5- Surface flow and velocity

Surface runoff in a watershed occurs first as a thin sheet of overland flow in the upper slopes for a short distance (< 100 ft) and then eventually produce or merge to the channel flow that you see in gullies, streams and rivers. The goal for this section is to briefly quantify the properties of overland sheet flow.
Figure 20: Overland flow to the streams and modeling concepts
Here p is precipitation intensity [L/T], θ is the surface slope, is infiltration rate [L/T], V is average velocity [L/T], L is slope length [L], is slope [L/L], and y is the flow depth [L].
Let's write a mass balance or continuity equation for the above control volume as follows:
Inflow: (precipitation)
Outflow: (infiltration)
Outflow: (discharge per unit width)
Conservation of mass results in =.
Note that in the above expression the unit is in terms of discharge per unit width []. Thus, the overland flow per unit width at the end of the slope is:
Recall: From the Darcy-Weisbach equation, we have
where f is the friction coefficient [-], L: pipe length [L]; D: pipe diameter [L]; V: Velocity []; and : head loss [L].
By definition, the hydraulic radius in pipe or channel is defined as:
A: wetter area [
P: wetted perimeter [
For example, for a rectangular channel we have : (when ). Recall that for a pipe and thus , when the the Darcy-Weisbach equation is used for a rectangular channel.
Laminar Overland Flow
Next we need to use the conservation of momentum to define the average velocity and height of the overland sheet flow. These equations depend on whether flow is laminar (parallel streamlines) or turbulent (mixed by eddies), which can be determined by the flow Reynold's number:
(pipe) (wide rectangular channel)
where ν is kinematic viscosity []. We know from fluid mechanics that generally the flow is laminar when . Recall that from the Moody diagram, we have for laminar flow in pipes; however, for uniform overland sheet flow, the following formula shall be used for laminar overland sheet flow (Chow 1998):
where p is the precipitation rate [].
Figure 21: Recall that the Energy Grade Line (EGL) is and for a uniform flow, we have , where is the slope of the EGL, is the head loss over the horizontal length L.
From the Darcy-Weisbach equation, for a uniform flow, and then we have
where .
Turbulent Overland Flow
Once overland flow becomes turbulent (), the roughness factorf becomes independent of the Reynolds number}, and the Manning's equation shall be used to explain the velocity of overland sheet flow as follows:
where n is Manning's roughness coefficient and is the hydraulic radius for a wide open channel. The Manning coefficient, is an empirically derived coefficient, which is dependent on many factors, including surface roughness and channel sinuosity.
Substitute into Manning's equation one can obtain:
Typical values for the Manning's coefficients n for overland flow are given on the next slide. These values are valid for sheet flow with depths less than 0.1 ft. We will revisit the Manning equation in the next chapters.
Figure 20: Typical values of the Manning roughness coefficient ($n$) for overland sheet flow.
Figure 20: Hydraulic radius formulas for simple open channel geometries. These formulas will be revisited in the next chapters.
Travel Time
One main reason for defining the velocity of these runoff mechanisms is to determine the travel time in a basin. As we know, for different segments of the river with the same velocity, the travel time can be approximated as follows:
Every point within a watershed will have a travel time based on its flowpath to the outlet. When rainfall begins, the areas with the shortest travel times are contributing first to the outflow. As the storm progresses, larger areas of the watershed contribute to the outflow until the time when all areas in the watershed are contributing to flow at the outlet. This time scale is known as the time of concentration ()} and can also be defined as the travel time of precipitation water from the longest flow path to the outlet.
Figure 24: Schematic showing the growing contributing areas and the time of concentration.
Table 1: Approximate average velocities in [ft/s] of runoff flow for calculating time of concentration. The unconcentrated condition addresses the flow velocity in the upper part of the watershed prior to the overland flows accumulating in a channel, while concentrated condition refers to the flow velocity in a channel network. It is important to note that the velocity values in concentrated condition are based on rough estimates of the Manning coefficient, which may vary with channel size and conditions (From drainage manual, Texas highway department, 1970).
Example: Calculate the time of concentration of a watershed in which, the longest flow path covers 100 ft of pasture at slope 5\% and a 1000 ft long rectangular channel having 2 ft width with Manning coefficient and slope of %. The channel receives a lateral flow of 0.0096 cfs/ft.
The flow velocity is 3 ft/s for the 100 ft of pasture with slope 5% (see Table 1) and thus . For the rectangular channel we have
The total discharge is and . Thus, after finding y, we can compute . We solve the problem for river segments 200 ft.
Figure 25: Computing the time of concentration for intervals of 200 [ft] for a channel of 1000 ft long. The depth of the flow $y$ in third row of the table is obtained by solving for the root of the Manning equation for a rectangular channel. The depth are calculated for the flow rate in the second row and ft. The represents the average velocity over each stream interval. For example, [ft/s] for the second interval.

6- Horton Laws of Stream Network

We will now briefly cover some basics of the stream network. One of the most common ways to organize the complex configuration of connected stream reaches in a river network is using Horton Stream Ordering. Basically, the channels furthest upstream that have no channels flowing into them are order 1. Then if two or more channels of a same order feed into a downstream reach, that reach becomes the next order, i.e., two order 1 streams feed into an order two streams. If two streams of different orders flow into another reach, that reach retains the highest order, i.e. an order 3 and order 1 stream feed into an order 3 stream.
Figure 26: Example of Horton stream ordering (from Hewlett and Nutter, 1969)
Horton and others found some significant scaling relationships from this ordering. The bifurcation ratio of the stream network is found through Horton's Law of Stream Number}:
where is the number of streams with order i. The bifurcation ratio, , has typical values ranging from 3 to 5. The next scaling law is Horton's Law of Stream Length:
where is the average length of streams with order i. Furthermore, one of the most important findings to hydrology is Horton's Law of Stream Contributing Area:
where is the average contributing area of streams with order i.
Figure 27: Example of Stream Laws from nested Mamon basins (Valdes et al., 1979).